Difference between revisions of "Cube (geometry)"
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| − | A '''cube''', or regular '''hexahedron''', is a solid composed of six [[Square (geometry) | square]] [[face]]s. A cube is dual to the regular [[octahedron]] and has [[octahedral symmetry]]. A cube is a [[Platonic solid]]. | + | A '''cube''', or regular '''hexahedron''', is a solid composed of six [[Square (geometry) | square]] [[face]]s. A cube is [[Platonic solid #Duality | dual]] to the regular [[octahedron]] and has [[octahedral symmetry]]. A cube is a [[Platonic solid]]. All edges of cubes are equal to each other. |
| + | |||
| + | The cube is also a square [[parallelepiped]], an equilateral cuboid, and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. | ||
| + | |||
==Formulas== | ==Formulas== | ||
A cube with [[edge]]-[[length]] <math>s</math> has: | A cube with [[edge]]-[[length]] <math>s</math> has: | ||
| − | * Four space [[diagonal]]s of | + | * Four space [[diagonal]]s of same lengths <math>s\sqrt{3}</math>(<math>\sqrt{s^2+s^2+s^2}=\sqrt{3s^2}=s\sqrt{3}</math>) |
| − | * [[Surface area]] <math>6s^2</math> | + | * [[Surface area]] of <math>6s^2</math>. (6 sides of areas <math>s \cdot s</math>.) |
| − | * [[Volume]] <math>s^3</math> | + | * [[Volume]] <math>s^3</math>(<math>s \cdot s \cdot s</math>) |
* A [[circumscribe]]d [[sphere]] of [[radius]] <math>\frac{s\sqrt{3}}{2}</math> | * A [[circumscribe]]d [[sphere]] of [[radius]] <math>\frac{s\sqrt{3}}{2}</math> | ||
* An [[inscribe]]d sphere of radius <math>\frac{s}{2}</math> | * An [[inscribe]]d sphere of radius <math>\frac{s}{2}</math> | ||
* A sphere [[tangent]] to all of its edges of radius <math>\frac{s\sqrt{2}}{2}</math> | * A sphere [[tangent]] to all of its edges of radius <math>\frac{s\sqrt{2}}{2}</math> | ||
| + | * A regular tetrahedron can fit in exactly two ways inside a cube | ||
| + | |||
| + | * For any cube whose circumscribing sphere has radius <math>R</math>, and for any given point in the its 3D dimensional space with distances <math>d_i</math> from the cube's eight vertices, we have: <cmath>\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.</cmath> | ||
==See also== | ==See also== | ||
Latest revision as of 11:04, 24 April 2023
A cube, or regular hexahedron, is a solid composed of six square faces. A cube is dual to the regular octahedron and has octahedral symmetry. A cube is a Platonic solid. All edges of cubes are equal to each other.
The cube is also a square parallelepiped, an equilateral cuboid, and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.
Formulas
has:
- Four space diagonals of same lengths
(
) - Surface area of
. (6 sides of areas 
.) - Volume
(
) - A circumscribed sphere of radius
- An inscribed sphere of radius
- A sphere tangent to all of its edges of radius
- A regular tetrahedron can fit in exactly two ways inside a cube
(
)
. (6 sides of areas 
.)
(
)
- For any cube whose circumscribing sphere has radius
, and for any given point in the its 3D dimensional space with distances 
from the cube's eight vertices, we have: ![\[\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.\]](//latex.artofproblemsolving.com/c/6/9/c690a5a8732c460805c5e0bc569e364904e1a785.png)
, and for any given point in the its 3D dimensional space with distances 
from the cube's eight vertices, we have: ![\[\frac{\sum_{i=1}^{8} d_i^2}{8} + \frac{16R^4}{9} = (\frac{\sum_{i=1}^{8} d_i^2}{8}+\frac{2R^2}{3})^2.\]](http://latex.artofproblemsolving.com/c/6/9/c690a5a8732c460805c5e0bc569e364904e1a785.png)
See also
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